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Popolare Algebra >

espandere (2x^2+a/(x^3))^{10}

Soluzione

espandere

(2 x^{2} + \frac{a}{x ^{3}})^{10}

Soluzione

1024 x^{20} + 5120 a x^{15} + 11520 a^{2} x^{10} + 15360 a^{3} x^{5} + 13440 a^{4} + \frac{8064 a ^{5}}{x ^{5}} + \frac{3360 a ^{6}}{x ^{10}} + \frac{960 a ^{7}}{x ^{15}} + \frac{180 a ^{8}}{x ^{20}} + \frac{20 a ^{9}}{x ^{25}} + \frac{a ^{10}}{x ^{30}}

Fasi della soluzione

(2 x^{2} + \frac{a}{x ^{3}})^{10}

Applicare il teorema binomiale:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = 2 x^{2}, b = \frac{a}{x ^{3}}

= i = 0 \sum 10 (i 10) (2 x^{2})^{(10 - i)} (\frac{a}{x ^{3}})^{i}

Espandere la sommatoria

= \frac{10 !}{0 ! ( 10 - 0 ) !} (2 x^{2})^{10} (\frac{a}{x ^{3}})^{0} + \frac{10 !}{1 ! ( 10 - 1 ) !} (2 x^{2})^{9} (\frac{a}{x ^{3}})^{1} + \frac{10 !}{2 ! ( 10 - 2 ) !} (2 x^{2})^{8} (\frac{a}{x ^{3}})^{2} + \frac{10 !}{3 ! ( 10 - 3 ) !} (2 x^{2})^{7} (\frac{a}{x ^{3}})^{3} + \frac{10 !}{4 ! ( 10 - 4 ) !} (2 x^{2})^{6} (\frac{a}{x ^{3}})^{4} + \frac{10 !}{5 ! ( 10 - 5 ) !} (2 x^{2})^{5} (\frac{a}{x ^{3}})^{5} + \frac{10 !}{6 ! ( 10 - 6 ) !} (2 x^{2})^{4} (\frac{a}{x ^{3}})^{6} + \frac{10 !}{7 ! ( 10 - 7 ) !} (2 x^{2})^{3} (\frac{a}{x ^{3}})^{7} + \frac{10 !}{8 ! ( 10 - 8 ) !} (2 x^{2})^{2} (\frac{a}{x ^{3}})^{8} + \frac{10 !}{9 ! ( 10 - 9 ) !} (2 x^{2})^{1} (\frac{a}{x ^{3}})^{9} + \frac{10 !}{10 ! ( 10 - 10 ) !} (2 x^{2})^{0} (\frac{a}{x ^{3}})^{10}

Semplificare

\frac{10 !}{0 ! ( 10 - 0 ) !} (2 x^{2})^{10} (\frac{a}{x ^{3}})^{0} : 1024 x^{20}

Semplificare

\frac{10 !}{1 ! ( 10 - 1 ) !} (2 x^{2})^{9} (\frac{a}{x ^{3}})^{1} : 5120 a x^{15}

Semplificare

\frac{10 !}{2 ! ( 10 - 2 ) !} (2 x^{2})^{8} (\frac{a}{x ^{3}})^{2} : 11520 a^{2} x^{10}

Semplificare

\frac{10 !}{3 ! ( 10 - 3 ) !} (2 x^{2})^{7} (\frac{a}{x ^{3}})^{3} : 15360 a^{3} x^{5}

Semplificare

\frac{10 !}{4 ! ( 10 - 4 ) !} (2 x^{2})^{6} (\frac{a}{x ^{3}})^{4} : 13440 a^{4}

Semplificare

\frac{10 !}{5 ! ( 10 - 5 ) !} (2 x^{2})^{5} (\frac{a}{x ^{3}})^{5} : \frac{8064 a ^{5}}{x ^{5}}

Semplificare

\frac{10 !}{6 ! ( 10 - 6 ) !} (2 x^{2})^{4} (\frac{a}{x ^{3}})^{6} : \frac{3360 a ^{6}}{x ^{10}}

Semplificare

\frac{10 !}{7 ! ( 10 - 7 ) !} (2 x^{2})^{3} (\frac{a}{x ^{3}})^{7} : \frac{960 a ^{7}}{x ^{15}}

Semplificare

\frac{10 !}{8 ! ( 10 - 8 ) !} (2 x^{2})^{2} (\frac{a}{x ^{3}})^{8} : \frac{180 a ^{8}}{x ^{20}}

Semplificare

\frac{10 !}{9 ! ( 10 - 9 ) !} (2 x^{2})^{1} (\frac{a}{x ^{3}})^{9} : \frac{20 a ^{9}}{x ^{25}}

Semplificare

\frac{10 !}{10 ! ( 10 - 10 ) !} (2 x^{2})^{0} (\frac{a}{x ^{3}})^{10} : \frac{a ^{10}}{x ^{30}}

= 1024 x^{20} + 5120 a x^{15} + 11520 a^{2} x^{10} + 15360 a^{3} x^{5} + 13440 a^{4} + \frac{8064 a ^{5}}{x ^{5}} + \frac{3360 a ^{6}}{x ^{10}} + \frac{960 a ^{7}}{x ^{15}} + \frac{180 a ^{8}}{x ^{20}} + \frac{20 a ^{9}}{x ^{25}} + \frac{a ^{10}}{x ^{30}}

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